ONT Re: Differential Logic
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DLOG. Note D59
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Remainder of Conjunction
| I bequeath myself to the dirt to grow from the grass I love,
| If you want me again look for me under your bootsoles.
|
| You will hardly know who I am or what I mean,
| But I shall be good health to you nevertheless,
| And filter and fibre your blood.
|
| Failing to fetch me at first keep encouraged,
| Missing me one place search another,
| I stop some where waiting for you
|
| Walt Whitman, 'Leaves of Grass', [Whi, 88]
Let us now recapitulate the story so far. In effect, we have
been carrying out a decomposition of the enlarged proposition
EJ in a series of stages. First, we considered the equation
EJ = !e!J + DJ, which was involved in the definition of DJ as
the difference EJ - !e!J. Next, we contemplated the equation
DJ = dJ + rJ, which expresses DJ in terms of two components,
the differential dJ that was just extracted and the residual
component rJ = DJ - dJ. This remaining proposition rJ can
be computed as shown in Table 47.
Table 47. Computation of rJ
o-------------------------------------------------------------------------------o
| |
| rJ = DJ + dJ |
| |
o-------------------------------------------------------------------------------o
| |
| DJ = u v ((du)(dv)) + u (v)(du) dv + (u) v du (dv) + (u)(v) du dv |
| |
| dj = u v (du, dv) + u (v) dv + (u) v du + (u)(v) . 0 |
| |
o-------------------------------------------------------------------------------o
| |
| rJ = u v du dv + u (v) du dv + (u) v du dv + (u)(v) du dv |
| |
o-------------------------------------------------------------------------------o
As it happens, the remainder rJ falls under the description
of a second order differential rJ = d^2.J. This means that
the expansion of EJ in the form:
EJ = !e!J + DJ
= !e!J + dJ + rJ
= d^0.J + d^1.J + d^2.J
which is nothing other than the propositional analogue
of a Taylor series, is a decomposition that terminates
in a finite number of steps.
Jon Awbrey
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